Extrude and snap to find the center (if you doesn'tdon't already have). Use Shift SShift+S > Cursor to selectedCursor to Selected to place the 3D cursorCursor there, then delete the center vertex.
Add a dense sphere and scale it so to fit exactly (use snap functions, expeciallyespecially using BB key to align vertex to vertex). Then enable Face snap with "Face Project"Face Project individual elements on.
Start building topology as desired.
When finished, select all newly created vertices, press Ctrl GCtrl+G > Assign to new groupAssign to New Group.
Add a Shrinkwrap modifierShrinkwrap Modifier, limited to the new Vertex Group, targeting the sphere. Right click and choose Smooth vertices Smooth Vertices (smoothing 1.0, repeat many times).
ApplyApply the Shrinkwrap Modifier (Ctrl ACtrl+A on the modifier).